General diffusion processes as the limit of time-space Markov chains
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866929589909979136 |
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| author | Anagnostakis, Alexis Lejay, Antoine Villemonais, Denis |
| author_facet | Anagnostakis, Alexis Lejay, Antoine Villemonais, Denis |
| contents | We prove the convergence of the law of grid-valued random walks, which can be seen as time-space Markov chains, to the law of a general diffusion process. This includes processes with sticky features, reflecting or absorbing boundaries and skew behavior. We prove that the convergence occurs at any rate strictly inferior to $(1/4) \wedge (1/p)$ in terms of the maximum cell size of the grid, for any $p$-Wasserstein distance. We also show that it is possible to achieve any rate strictly inferior to $(1/2) \wedge (2/p)$ if the grid is adapted to the speed measure of the diffusion, which is optimal for $p\le 4 $. This result allows us to set up asymptotically optimal approximation schemes for general diffusion processes. Last, we experiment numerically on diffusions that exhibit various features. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_03713 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | General diffusion processes as the limit of time-space Markov chains Anagnostakis, Alexis Lejay, Antoine Villemonais, Denis Probability 60J60, 60F17 We prove the convergence of the law of grid-valued random walks, which can be seen as time-space Markov chains, to the law of a general diffusion process. This includes processes with sticky features, reflecting or absorbing boundaries and skew behavior. We prove that the convergence occurs at any rate strictly inferior to $(1/4) \wedge (1/p)$ in terms of the maximum cell size of the grid, for any $p$-Wasserstein distance. We also show that it is possible to achieve any rate strictly inferior to $(1/2) \wedge (2/p)$ if the grid is adapted to the speed measure of the diffusion, which is optimal for $p\le 4 $. This result allows us to set up asymptotically optimal approximation schemes for general diffusion processes. Last, we experiment numerically on diffusions that exhibit various features. |
| title | General diffusion processes as the limit of time-space Markov chains |
| topic | Probability 60J60, 60F17 |
| url | https://arxiv.org/abs/2206.03713 |